Question

What is $\sqrt{106}$ in simplified radical form?

Answer 1

Hint: We must first find the prime factorisation of the number inside the radical sign, and then look for the prime numbers that are appearing at least twice. We can then bring such prime numbers outside the radical sign to get the simplified radical form.

Complete step by step solution:
We are aware that a radical is nothing but the mathematical opposite of exponents. A square root is the smallest radical. Any expression that contains a root in it, is called a radical expression. The expression or number, which is inside the root symbol is called a radicand. And we must also note that the small number, written in front of the root symbol, is called the index of the radical.
We know that a simplified radical form is a method of representation of radicals in which the number inside the radical symbol does not contain any square, or cube of any number, as per the index of that radical.
Also, for square roots, we have index = 2.
Thus, we now know that to be in simplified radical form, the number inside the radical sign must not contain squares of any number.
In this problem, we have to express $\sqrt{106}$ in simplified radical form.
First, let us find the prime factorization of 106.
$\begin{align}
  & \text{ 2}\left| \!{\underline {\,
  106 \,}} \right. \\
 & 53\left| \!{\underline {\,
  53 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence, we have $106=2\times 53$ .
We can see that, in the prime factorisation, each prime number occurs just once.
So, from this prime factorisation, we can say that 106 has no squares within.
Thus, we can very easily say that $\sqrt{106}$ is already in its simplified radical form.
Or, we can say that the simplified radical form of $\sqrt{106}$ is $\sqrt{106}$.

Note: We must be very clear with the terminologies used in the context of the radicals. We must also keep in mind that the prime factorization of each number is unique. Thus, it is totally fine to comment about the number by analysing its prime factorization.